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Which of the following is closest to the difference between sum of all

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Which of the following is closest to the difference between sum of all [#permalink]

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New post 07 Sep 2016, 12:04
4
00:00
A
B
C
D
E

Difficulty:

  95% (hard)

Question Stats:

39% (01:11) correct 61% (01:49) wrong based on 64 sessions

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Which of the following is closest to the difference between sum of all proper fractions (fractions less than 1) in the form 1/x , where x is a positive digit, and the product of all proper fractions in the form y/(y+1), where y is a positive digit?
A) 2.82
B) 2.72
C) 1.82
D) 1.72
E) 0.82
Spoiler: OA

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Which of the following is closest to the difference between sum of all [#permalink]

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New post 07 Sep 2016, 16:24
First term = 1/2 +1/3 +1/4 +1/5 +1/6 +1/7 +1/8 + 1/9

Second term = 1/2 * 2/3 * 3/4 * 4/5 * 5/6 * 6/7 * 7/8 * 8/9 * 9/10 = 1/10

First term - Second term = 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 +1/8 + 1/9 - 1/10

Now rearrange the above equation for simplification:

= (1/2 + 1/5 - 1/10) +( 1/3 + 1/6 + 1/9) + (1/4 + +1/8) +1/7

= 6/10 + 11/18 + 3/8 +1/7

6/10 , 11/18, 3/8 are approximately equal to 0.5

=0.5+0.5+0.5+0.14 = 1.64

The closest answer choice is D

Alternate approach:

We know that answer = First term - 0.1

Answer choices have pairs: A & B and C & D are two pairs

2.82 - 0.1 = 2.72
1.82 - 0.1 = 1.72

So the answer choice has to be either B or D

Pick either of two.

Note: Please use the second method only if your are running out of time.
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Which of the following is closest to the difference between sum of all [#permalink]

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New post 07 Sep 2016, 18:34
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Sum of all proper fractions (fractions less than 1) in the form 1/x, where x is a positive digit:

1/1 + 1/2 + 1/3 +.....+ 1/9
This is a harmonic progression. Harmonic progression is inverse of arithmetic progression.
Approximate sum of a harmonic progression with even number of terms = number of terms * (average of middle 2 terms)
Approximate sum of a harmonic progression with odd number of terms = number of terms * (middle term)
The actual sum will be slightly more than the approximation.

Here we have 9 terms (odd).
Sum = 9 * 1/5
= 9/5

Product of all proper fractions in the form y/(y+1), where y is a positive digit:

1/2 * 2/3 * 3/4 *.....*9/10
We will be left with 1/10.

Required = 9/5 - 1/10
= 1.8 - 0.1
= 1.7

Closest is 1.72

Answer (D).
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Re: Which of the following is closest to the difference between sum of all [#permalink]

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New post 07 Sep 2016, 18:57
Is something missing in the question stem?? something like for all n>1 and n<10?
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Re: Which of the following is closest to the difference between sum of all [#permalink]

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New post 07 Sep 2016, 18:59
x and y are positive digits (1 to 9).


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Re: Which of the following is closest to the difference between sum of all [#permalink]

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New post 07 Sep 2016, 19:02
Argo wrote:
Is something missing in the question stem?? something like for all n>1 and n<10?


Agreed.
The word digit is a bit unclear but I think it means digits 1 to 9.
0 can be excluded because as per the second part, we would need to do multiple by 1/0 which is undefined.
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Re: Which of the following is closest to the difference between sum of all [#permalink]

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New post 07 Sep 2016, 19:05
1
Donnie84 wrote:
Argo wrote:
Is something missing in the question stem?? something like for all n>1 and n<10?


Agreed.
The word digit is a bit unclear but I think it means digits 1 to 9.
0 can be excluded because as per the second part, we would need to do multiple by 1/0 which is undefined.



0 is neither positive nor negative. So we can exclude that since we are considering positive digits.
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Re: Which of the following is closest to the difference between sum of all [#permalink]

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New post 13 Aug 2017, 23:27
1
duahsolo wrote:
Which of the following is closest to the difference between sum of all proper fractions (fractions less than 1) in the form 1/x , where x is a positive digit, and the product of all proper fractions in the form y/(y+1), where y is a positive digit?
A) 2.82
B) 2.72
C) 1.82
D) 1.72
E) 0.82


The question is a bit tricky because of the word digit. Most people assume it to be numbers and are do calculations for values of x upto infinity..

So, 1st term = 1/2 +1/3+1/4+1/5+1/6+1/7+1/8 +1/9
=(1/2+1/4+1/8)+(1/3+1/6+1/9)+(1/5+1/7)
=(0.500+0.250+0.125) +(0.333+0.167 +0.111)+(0.200+0.143)
= 0.875 +0.611+0.343
=1.829

2nd term = 1/2*2/3*....*8/9 = 1/9 = 0.111

So, 1st term - 2nd term = 1.829 - 0.111 = 1.718 ~ 1.72

Answer D
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Re: Which of the following is closest to the difference between sum of all   [#permalink] 13 Aug 2017, 23:27
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